Thin-film equation

In fluid mechanics, the thin-film equation is a partial differential equation that approximately predicts the time evolution of the thickness $h$ of a liquid film that lies on a surface. The equation is derived via lubrication theory which is based on the assumption that the length-scales in the surface directions are significantly larger than in the direction normal to the surface. In the non-dimensional form of the Navier-Stokes equation the requirement is that terms of order $ε2$ and $ε2Re$ are negligible, where $ε ≪ 1$ is the aspect ratio and $Re$ is the Reynolds number. This significantly simplifies the governing equations. However, lubrication theory, as the name suggests, is typically derived for flow between two solid surfaces, hence the liquid forms a lubricating layer. The thin-film equation holds when there is a single free surface. With two free surfaces, the flow must be treated as a viscous sheet.

Definition
The basic form of a 2-dimensional thin film equation is

$$\frac{\partial h}{\partial t} = -\nabla\cdot \mathbf{Q}$$

where the fluid flux $$\mathbf{Q}$$ is

$$\mathbf{Q} = \frac{h^3}{3\mu } \left[\nabla \right(\gamma \nabla^2 h + \rho \mathbf{g} \cdot \mathbf{\hat{e}_n}) + \rho \mathbf{g} \cdot \mathbf{\hat{e}_i}] + \frac{h^2}{2\mu} \mathbf{A} $$ ,

and &mu; is the viscosity (or dynamic viscosity) of the liquid, h(x,y,t) is film thickness, &gamma; is the interfacial tension between the liquid and the gas phase above it, $$\rho$$ is the liquid density and $$\mathbf{A}$$ the surface shear. The surface shear could be caused by flow of the overlying gas or surface tension gradients. The vectors $$\mathbf{\hat{e}_i}$$ represent the unit vector in the surface co-ordinate directions, the dot product serving to identify the gravity component in each direction. The vector $$\mathbf{\hat{e}_n}$$ is the unit vector perpendicular to the surface.

A generalised thin film equation is discussed in


 * $$\frac{\partial h}{\partial t} = -\frac 1 {3\mu} \nabla\cdot \left( h^n \, \nabla \left( \gamma \, \nabla^2 h \right)\right)$$.

When $$n<3$$ this may represent flow with slip at the solid surface whole $$n=1$$ describes the thickness of a thin bridge between two masses of fluid in a Hele-Shaw cell. The value $$n=3$$ represents surface tension driven flow.

A form frequently investigated with regard to the rupture of thin liquid films involves  the addition of a disjoining pressure &Pi;(h) in the equation, as in


 * $$\frac{\partial h}{\partial t} = -\frac 1 {3\mu} \nabla\cdot\left(h^3 \nabla \left( \gamma \, \nabla^2 h-\Pi (h) \right)\right)$$

where the function &Pi;(h) is usually very small in value for moderate-large film thicknesses h and grows very rapidly when h goes very close to zero.

Properties
Physical applications, properties and solution behaviour of the thin-film equation are reviewed in. With the inclusion of phase change at the substrate a form of thin film equation for an arbitrary surface is derived in. A detailed study of the steady-flow of a thin film near a moving contact line is given in. For a yield-stress fluid flow driven by gravity and surface tension is investigated in.

For purely surface tension driven flow it is easy to see that one static (time-independent) solution is a paraboloid of revolution


 * $$h(x,y) = A - B(x^2 + y^2 ) \, $$

and this is consistent with the experimentally observed spherical cap shape of a static sessile drop, as a "flat" spherical cap that has small height can be accurately approximated in second order with a paraboloid. This, however, does not handle correctly the circumference of the droplet where the value of the function h(x,y) drops to zero and below, as a real physical liquid film can't have a negative thickness. This is one reason why the disjoining pressure term &Pi;(h) is important in the theory.

One possible realistic form of the disjoining pressure term is


 * $$\Pi (h) = B\left[\left(\frac{h_*} h \right)^n - \left(\frac{h_*} h \right)^m \right]$$

where B, h*, m and n are some parameters. These constants and the surface tension $$\gamma$$ can be approximately related to the equilibrium liquid-solid contact angle $$\theta_e$$ through the equation


 * $$B \approx \frac{(m-1)(n-1)}{h_* (n-m)}\gamma (1-\cos \theta_e ) $$.

The thin film equation can be used to simulate several behaviors of liquids, such as the fingering instability in gravity driven flow.

The lack of a second-order time derivative in the thin-film equation is a result of the assumption of small Reynold's number in its derivation, which allows the ignoring of inertial terms dependent on fluid density $$\rho$$. This is somewhat similar to the situation with Washburn's equation, which describes the capillarity-driven flow of a liquid in a thin tube.